Mean time to lose lock for the "Langevin"-type delay-locked loop

Welti, A.L.; Bobrovsky, B.Z.

doi:10.1109/26.310610

Cited by 16 publications

(21 citation statements)

References 16 publications

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“…Loop equation normalisation. Following References [13,14,16,17], we scale the time, t ¼ t Ã =g 0 . The normalised equation is preferred since only in this equation one can properly compare the noise intensity with the restoring force.…”

Section: Mtll Calculationmentioning

confidence: 99%

Optimal design of an all-digital chip timing recovery loop for direct-sequence spread-spectrum systems

Hasson

Bobrovsky

2005

Eur. Trans. Telecomm.

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SUMMARYNoncoherent digital delay lock loops (DDLL) are suited for chip timing synchronisation in band-limited direct-sequence spread-spectrum (DSSS) demodulators. The diffusion approximation and the singular perturbation method are used in this paper to calculate the mean time to lose lock (MTLL) of the DDLL. Loop bandwidth optimisation for first order loop with severe Doppler is presented. A simple design rule for the loop bandwidth and a systematic approach for the loop threshold calculation are presented.

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Section: Mtll Calculationmentioning

confidence: 99%

Optimal design of an all-digital chip timing recovery loop for direct-sequence spread-spectrum systems

Hasson

Bobrovsky

2005

Eur. Trans. Telecomm.

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“…Figure shows the first‐order DLL equivalent model with the constant loop filter F ( s ) = K . Also, Figure introduces the timing error normalised with respect to the chip duration

bold-italicϵ (t) MathClass-rel= \frac{T (t) MathClass-bin- \overset{normalT}{true} (t)}{T_{normalC}}

The normalised loop discriminator in Figure

S (ϵ) MathClass-rel= \frac{R_{normalc} ((), ϵ MathClass-bin- \frac{δ}{2}) MathClass-bin- R_{normalc} ((), ϵ MathClass-bin+ \frac{δ}{2})}{\underset{normalmax}{msub} (||, R_{normalc} ((), ϵ MathClass-bin- \frac{δ}{2}) MathClass-bin- R_{normalc} ((), ϵ MathClass-bin+ \frac{δ}{2}))}

is based on the autocorrelation function of the pseudo‐noise code R c ( τ ) = E [ c * ( t ) c ( t + τ )]. In Equation , δ denotes the correlator spacing between the early correlation

R_{normalc} ((), ϵ MathClass-bin- \frac{δ}{2})

and the late correlation

R_{normalc} ((), ϵ MathClass-bin+ \frac{δ}{2})

.…”

Section: Analysis Of First‐order Delay‐locked Loopsmentioning

confidence: 99%

“…The SDEs in Figure read

\begin{matrix} aligned \\ right z MathClass-open(t MathClass-close) & left = - K \sqrt{P} S MathClass-open(ϵ MathClass-open(t MathClass-close) MathClass-close) - K \sqrt{N_{0}} \overset{MathClass-op̃}{n} MathClass-open(t MathClass-close), & right \\ right ϵ ° MathClass-open(t MathClass-close) & left = \frac{v_{0}}{c_{0}} R_{C} - z MathClass-open(t MathClass-close) \end{matrix}

where the two‐sided PSD of the additive white Gaussian noise

\overset{normaln}{true} (t)

is 1 W/Hz. With the loop offset

u MathClass-rel= \frac{v_{0}}{c_{0}} R_{normalC}

the first‐order SDE corresponding to Equation reads

\begin{matrix} aligned \\ right ϵ ° MathClass-open(t MathClass-close) & left = u - K \sqrt{P} S MathClass-open(ϵ MathClass-open(t MathClass-close) MathClass-close) - K \sqrt{N_{0}} \overset{MathClass-op̃}{n} MathClass-open(t MathClass-close) & right \\ right & left = u - K \sqrt{P} S ° MathClass-open(0 MathClass-close) ϵ MathClass-open(t MathClass-cl... \end{matrix}

…”

Section: Analysis Of First‐order Delay‐locked Loopsmentioning

confidence: 99%

“…The single‐pole loop filter F ( s ) = K ∕ ( sτ + 1) and the other definitions in Figure yield the SDEs

\begin{matrix} aligned \\ right z ° MathClass-open(t MathClass-close) & left = - \frac{z MathClass-open(t MathClass-close)}{τ} + \frac{K \sqrt{P}}{τ} S (ϵ MathClass-open(t MathClass-close)) + \frac{K}{τ} \sqrt{N_{0}} \overset{MathClass-op̃}{n} MathClass-open(t MathClass-close) & right \\ right ϵ ° MathClass-open(t MathClass-close) & left = \frac{T ° MathClass-open(t MathClass-close)}{T_{C}} - \frac{\overset{MathClass-op̂}{T} ° MathClass-open(t MathClass-close)}{T_{C}} = \frac{v_{0}}{c_{0}} R_{C} - z MathClass-open(t MathClass-close) \end{matrix}

where the noise process

\overset{normaln}{true} (t)

has the two‐sided PSD 1 W/Hz. The Langevin SDE follows from Equation as

{bold-italicϵ}^{MathClass-bin\circ MathClass-bin\circ} ((), t^{MathClass-bin*}) MathClass-bin+ β bold-italicϵ ° ((), t^{MathClass-bin*}) MathClass-bin+ normalS ((), bold-italicϵ ((), t^{MathClass-bin*})) MathClass-bin- u MathClass-rel= \sqrt{2}

…”

Section: Analysis Of Second‐order Langevin Delay‐locked Loopsmentioning

confidence: 99%

“…The first‐order SDE system after the time transformation

t^{MathClass-bin*} MathClass-rel= \overset{K}{true} t

reads

\begin{matrix} aligned \\ right ϵ ° (t^{*}) & left = - β y (t^{*}) - S (ϵ (t^{*})) - \sqrt{2 ρ} n (t^{*}) & right \\ right y ° (t^{*}) & left = - a + S (ϵ (t^{*})) + \sqrt{2 ρ} n (t^{*}) \end{matrix}

with the loop offset a and the auxiliary state variable

normaly (t) MathClass-rel= \frac{normalz (t)}{K_{1}}

and where the properties of the Brownian motion

\overset{normaln}{true} (t) MathClass-rel= \sqrt{\overset{K}{true}} normaln ((), t^{MathClass-bin*})

are used.…”

Section: Analysis Of General Second‐order Delay‐locked Loopsmentioning

confidence: 99%

See 2 more Smart Citations

A novel tracking jitter computation method for delay‐locked loops in spread‐spectrum systems

Groh

Gentner

Selva

2012

Trans. Emerging Tel. Tech.

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In spread‐spectrum navigation systems, the positioning error (or tracking accuracy) is determined from the jitter deviation of delay‐locked loops (DLLs). However, jitter simulations are computationally complex, and conventional analytical methods provide only unsatisfactory and incomplete jitter results. Thus, our contribution presents a novel method for the analytical jitter computation by mapping the stochastic differential equation (SDE) of the DLL onto an Ornstein–Uhlenbeck (OU) SDE. Its solution is the well‐known OU random process, which is a time‐variant Gaussian distribution. The expectation value and the variance of the OU random process yield the jitter deviation. Thus, we derive the analytical time‐variant jitter function with its transient response, which was to date unavailable. Contrary to previous jitter computations based on a loop transfer function, our method covers DLLs of any order. We obtain the loop parameters that minimise the jitter deviation analytically without computationally complex simulations. Above all, our method based on OU random processes enables for the first time an efficient joint analytical mean time to lose lock (MTLL) and jitter computation. Therefore, both computationally complex MTLL and jitter simulations become obsolete for many kinds of DLLs.Copyright © 2012 John Wiley & Sons, Ltd.

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Stability requirements of oscillators used for synchronization loops in communication systems

Welti¹,

Bernhard

1996

Trans Emerging Tel Tech

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Abstract. An imperfect clock timing recovery in receivers of modern communication systems degrades the performance of synchronization loops and, hence, the overall system reliability and data transmission quality. In this paper, the phase-locked loop (PLL) for the carrier recovery and the delay-locked loop (DLL) for the clock timing recovery of the pseudonoise (PN) code in directsequence spread-spectrum systems are analyzed regarding the impacts of imperfect oscillators (including phase noise and a frequency offset between transmitter and receiver) and channel noise on the mean exit time. The optimal design considerations to achieve a maximum mean exit time for second-order synchronization loops can be used to find the required oscillator stability, Le., the maximum tolerable frequency offset and oscillator phase noise.

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Mean time to lose lock for the "Langevin"-type delay-locked loop

Cited by 16 publications

References 16 publications

Optimal design of an all-digital chip timing recovery loop for direct-sequence spread-spectrum systems

Optimal design of an all-digital chip timing recovery loop for direct-sequence spread-spectrum systems

A novel tracking jitter computation method for delay‐locked loops in spread‐spectrum systems

Stability requirements of oscillators used for synchronization loops in communication systems

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